On a non-local free boundary problem modeling cell polarization

Abstract

The amplification of an external signal is a key step in direction sensing of biological cells, a process that contributes significantly in the regulation of cell shape. This thesis is concerned with a simple model for cell polarization as a response to a time-depending signal, which was previously proposed by Niethammer, Röger and Velázquez (2020). <br /> The model consists of a bulk-surface reaction-diffusion system of partial differential equations for different variants of a protein on the cell surface and interior respectively. The coupling is by a nonlinear Robin-type boundary condition for the bulk variable and a corresponding source term on the cell surface. We study solutions of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. <br /> It turns out that in suitable scaling limits solutions converge to solutions of some obstacle type problems. A distinguishing feature of these limiting problems is the presence of a term that depends in a non-local way on the support of the solution itself and makes the analysis quite challenging. <br /> First, we justify the well-posedness of these obstacle type problems. Moreover, we show an L1-contraction property of solutions, by means of which, we further prove that the steady states are globally stable. It is worth pointing out that, this first part of the thesis complements to a certain extent the former analysis while it also provides a more advanced insight on the limiting problems through this innovative L1-contraction property. <br /> In the second part of this thesis, we investigate qualitative properties of the free boundary. We conclude that there are necessary and sufficient conditions for the initial data that imply continuity of the support at t = 0. If one of these assumptions fail, then jumps of the support take place. We provide a complete characterization of the jumps for a large class of initial data as well. The continuity results concerning the set {u(·, t) > 0} can be further improved by imposing some additional assumptions on the initial data. In fact, restricting our analysis to the case of the unit sphere, we prove global in time continuity for the support of the solution. <br /> The latter part of this thesis allows for a better understanding of the evolution of the special non-local term that is involved in these limiting problems and depends on the set {u(·, t) > 0}, which can be useful for future analysis.